Existence of random invariant periodic curves via random semiuniform ergodic theorem

TL;DR

This paper extends ergodic theory to random dynamical systems, proving that dissipative systems possess invariant compact sets composed of finitely many random periodic curves, advancing understanding of their long-term behavior.

Contribution

It introduces a novel extension of ergodic theory to the random setting, demonstrating the existence of random periodic solutions in dissipative systems.

Findings

Random invariant compact sets can be decomposed into finite unions of random periodic curves.

The approach builds on and extends recent work by Zhao and Zheng.

The results provide new insights into the structure of solutions in random dynamical systems.

Abstract

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].